A002572 -id:A002572 - cp's OEIS frontend

A176485 First column of triangle in A176452.

1, 1, 1, 2, 4, 7, 13, 25, 48, 92, 176, 338, 649, 1246, 2392, 4594, 8823, 16945, 32545, 62509, 120060, 230598, 442910, 850701, 1633948, 3138339, 6027842, 11577747, 22237515, 42711863, 82037200, 157569867, 302646401, 581296715, 1116503866, 2144482948, 4118935248, 7911290530

Offset: 1

N. J. A. Sloane, Dec 07 2010

nonn

a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 3*p(k+1), see example. [Joerg Arndt, Dec 18 2012]

Row 2 of Table 1 of Elsholtz, row 1 being A002572. - Jonathan Vos Post, Aug 30 2011

			From _Joerg Arndt_, Dec 18 2012: (Start)
There are a(7+1)=25 compositions 7=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 3*p(k+1):
[ 1]  [ 1 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 1 2 ]
[ 3]  [ 1 1 1 1 2 1 ]
[ 4]  [ 1 1 1 1 3 ]
[ 5]  [ 1 1 1 2 1 1 ]
[ 6]  [ 1 1 1 2 2 ]
[ 7]  [ 1 1 1 3 1 ]
[ 8]  [ 1 1 2 1 1 1 ]
[ 9]  [ 1 1 2 1 2 ]
[10]  [ 1 1 2 2 1 ]
[11]  [ 1 1 2 3 ]
[12]  [ 1 1 3 1 1 ]
[13]  [ 1 1 3 2 ]
[14]  [ 1 2 1 1 1 1 ]
[15]  [ 1 2 1 1 2 ]
[16]  [ 1 2 1 2 1 ]
[17]  [ 1 2 1 3 ]
[18]  [ 1 2 2 1 1 ]
[19]  [ 1 2 2 2 ]
[20]  [ 1 2 3 1 ]
[21]  [ 1 2 4 ]
[22]  [ 1 3 1 1 1 ]
[23]  [ 1 3 1 2 ]
[24]  [ 1 3 2 1 ]
[25]  [ 1 3 3 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..2000
Christian Elsholtz, Clemens Heuberger, Daniel Krenn, Algorithmic counting of nonequivalent compact Huffman codes, arXiv:1901.11343 [math.CO], 2019.
Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, The number of Huffman codes, compact trees, and sums of unit fractions, arXiv:1108.5964 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.

Cf. A176452, A002572, A176503, A294775.

b[n_, r_, k_] := b[n, r, k] = If[n < r, 0, If[r == 0, If[n == 0, 1, 0], Sum[b[n-j, k*(r-j), k], {j, 0, Min[n, r]}]]];
a[n_] := b[2n-1, 1, 3];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

/* g.f. as given in the Elsholtz/Heuberger/Prodinger reference */
N=66;  q='q+O('q^N);
t=3;  /* t-ary: t=2 for A002572, t=3 for A176485, t=4 for A176503  */
L=2 + 2*ceil( log(N) / log(t) );
f(k) = (1-t^k)/(1-t);
la(j) = prod(i=1, j, q^f(i) / ( 1 - q^f(i) ) );
nm=sum(j=0, L, (-1)^j * q^f(j) * la(j) );
dn=sum(j=0, L, (-1)^j * la(j) );
gf = nm / dn;
Vec( gf )
/* Joerg Arndt, Dec 27 2012 */

a(n) = A294775(n-1,2). - Alois P. Heinz, Nov 08 2017

Extended by Jonathan Vos Post, Aug 30 2011

Added terms beyond a(20)=62509, Joerg Arndt, Dec 18 2012.

A176503 Leading column of triangle in A176463.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 15, 29, 57, 112, 220, 432, 848, 1666, 3273, 6430, 12632, 24816, 48754, 95783, 188177, 369696, 726312, 1426930, 2803381, 5507590, 10820345, 21257915, 41763825, 82050242, 161197933, 316693445, 622183778, 1222357651, 2401474098, 4717995460

Offset: 1

N. J. A. Sloane, Dec 07 2010

nonn

a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 4*p(k+1), see example. [Joerg Arndt, Dec 18 2012]

			From _Joerg Arndt_, Dec 18 2012: (Start)
There are a(6+1)=15 compositions 6=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 4*p(k+1):
[ 1]  [ 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 2 ]
[ 3]  [ 1 1 1 2 1 ]
[ 4]  [ 1 1 1 3 ]
[ 5]  [ 1 1 2 1 1 ]
[ 6]  [ 1 1 2 2 ]
[ 7]  [ 1 1 3 1 ]
[ 8]  [ 1 1 4 ]
[ 9]  [ 1 2 1 1 1 ]
[10]  [ 1 2 1 2 ]
[11]  [ 1 2 2 1 ]
[12]  [ 1 2 3 ]
[13]  [ 1 3 1 1 ]
[14]  [ 1 3 2 ]
[15]  [ 1 4 1 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Christian Elsholtz, Clemens Heuberger, Daniel Krenn, Algorithmic counting of nonequivalent compact Huffman codes, arXiv:1901.11343 [math.CO], 2019.
Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, The number of Huffman codes, compact trees, and sums of unit fractions, arXiv:1108.5964v1 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.

Cf. A176463, A194628 - A194633, A294775.

b[n_, r_, k_] := b[n, r, k] = If[n < r, 0, If[r == 0, If[n == 0, 1, 0], Sum[b[n-j, k*(r-j), k], {j, 0, Min[n, r]}]]];
a[n_] := b[3n-2, 1, 4];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

/* g.f. as given in the Elsholtz/Heuberger/Prodinger reference */
N=66;  q='q+O('q^N);
t=4;  /* t-ary: t=2 for A002572, t=3 for A176485, t=4 for A176503  */
L=2 + 2*ceil( log(N) / log(t) );
f(k) = (1-t^k)/(1-t);
la(j) = prod(i=1, j, q^f(i) / ( 1 - q^f(i) ) );
nm=sum(j=0, L, (-1)^j * q^f(j) * la(j) );
dn=sum(j=0, L, (-1)^j * la(j) );
gf = nm / dn;
Vec( gf )
/* Joerg Arndt, Dec 27 2012 */

a(n) = A294775(n-1,3). - Alois P. Heinz, Nov 08 2017

Added terms beyond a(13)=848, Joerg Arndt, Dec 18 2012

A002848 Number of maximal collections of pairwise disjoint subsets {X,Y,Z} of {1, 2, ..., n} with X + Y = Z (as in A002849), with the property that n is in one of the subsets.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 7, 15, 12, 30, 8, 32, 164, 21, 114, 867, 3226, 720, 4414, 24412, 4079, 31454, 3040, 25737, 252727, 20505, 191778, 2140186, 14554796, 1669221, 17754992, 148553131, 14708525, 177117401, 10567748, 138584026, 1953134982, 103372655, 1431596750, 22374792451, 218018425976, 16852166906, 254094892254

Offset: 0

N. J. A. Sloane

nonn

			Examples from _Alois P. Heinz_, Feb 12 2010:
A002848(7) = 3:
  [1, 3, 4], [2, 5, 7]
  [1, 5, 6], [3, 4, 7]
  [2, 3, 5], [1, 6, 7]
A002848(8) = 7:
  [1, 3, 4], [2, 6, 8]
  [1, 4, 5], [2, 6, 8]
  [1, 6, 7], [3, 5, 8]
  [2, 3, 5], [1, 7, 8]
  [2, 4, 6], [1, 7, 8]
  [2, 4, 6], [3, 5, 8]
  [3, 4, 7], [2, 6, 8]
A002848(10) = 12:
  [1, 4, 5], [2, 6, 8], [3, 7, 10]
  [1, 4, 5], [3, 6, 9], [2, 8, 10]
  [1, 5, 6], [3, 4, 7], [2, 8, 10]
  [1, 6, 7], [4, 5, 9], [2, 8, 10]
  [1, 7, 8], [2, 3, 5], [4, 6, 10]
  [1, 8, 9], [2, 3, 5], [4, 6, 10]
  [1, 8, 9], [2, 4, 6], [3, 7, 10]
  [1, 8, 9], [2, 5, 7], [4, 6, 10]
  [2, 4, 6], [3, 5, 8], [1, 9, 10]
  [2, 6, 8], [3, 4, 7], [1, 9, 10]
  [2, 6, 8], [4, 5, 9], [3, 7, 10]
  [2, 7, 9], [3, 5, 8], [4, 6, 10]
See A002849 for further examples.

R. K. Guy, "Sedlacek's Conjecture on Disjoint Solutions of x+y= z," in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, "Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics," in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.
Richard K. Guy, The unity of combinatorics, in Proc. 25th Iran. Math. Conf., Tehran, (1994), Math. Appl. 329 (1994) 129-159, Kluwer Acad. Publ., Dordrecht, 1995.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Martin Fuller, Table of n, a(n) for n = 0..52
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971. [Annotated scanned copy, with permission]
Christian Hercher and Frank Niedermeyer, Efficient Calculation the Number of Partitions of the Set {1, 2, ..., 3n} into Subsets {x, y, z} Satisfying x + y = z, arXiv:2307.00303 [math.CO], 2023.
Nigel Martin, Solving a conjecture of Sedlacek: maximal edge sets in the 3-uniform sumset hypergraphs, Discrete Mathematics, Volume 125, 1994, pp. 273-277.

Cf. A002849, A108235, A161826.

For n >= 2, a(n) = A002849(n) if n == 0,3,7,10 (mod 12), otherwise a(n) = A002849(n) - A002849(n-1). - _Franklin T. Adams-Watters; corrected by Max Alekseyev, Jul 06 2023

Edited by N. J. A. Sloane, Feb 10 2010, based on posting to the Sequence Fans Mailing List by Franklin T. Adams-Watters, R. K. Guy, R. H. Hardin, Alois P. Heinz, Andrew Weimholt, Max Alekseyev and others
a(32)-a(39) from Max Alekseyev, Feb 23 2012
Definition corrected by Max Alekseyev, Nov 16 2012, Jul 06 2023
a(40)-a(42) from Fausto A. C. Cariboni, Mar 12 2017
a(43)-a(44) computed from A002849 by Max Alekseyev, Jul 06 2023

A202705 Number of irreducible ways to split 1, 2, 3, ..., 3n into n arithmetic progressions each with 3 terms.

Original entry on oeis.org

1, 1, 1, 2, 6, 25, 115, 649, 4046, 29674, 228030, 1987700, 18402704, 188255116, 2030067605, 23829298479, 293949166112, 3909410101509, 54360507919179, 806312701922676

Offset: 0

N. J. A. Sloane, Dec 26 2011

"Irreducible" means that there is no j such that the first j of the triples are a partition of 1, ..., 3j.

R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "K".
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.] Gives a(0)-a(10).

All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.

A279197 Number of self-conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

Original entry on oeis.org

1, 1, 2, 2, 11, 11, 55, 58, 486, 442, 4218, 3924, 45096, 42013, 538537, 505830, 7368091, 6959545, 111877294, 105723374, 1886636688, 1763443165, 34585786729, 32401780965, 687085545694, 642233156868, 14691047314846, 13788837896728, 340221989868538, 317342350394678, 8327884506579315

Offset: 1

N. J. A. Sloane, Dec 15 2016

nonn

In Richard Guy's letter, the term 50 is marked with a question mark. Peter Kagey has shown that the value should be 55. - N. J. A. Sloane, Feb 15 2017
From Peter Kagey, Feb 14 2017: (Start)
An inseparable solution is one in which "there is no j such that the first j of the triples are a partition of 1, ..., 3j" (See A202705.)
A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.
(End)

			Examples of solutions X,Y,Z for n=5:
  2,4,3
  5,7,6
  1,15,8
  9,11,10
  12,14,13
and in his letter Richard Guy has drawn links pairing the first and fifth solutions, and the second and fourth solutions.
For n = 2 the a(2) = 1 solution is
  [(2,6,4),(1,5,3)].
For n = 3 the a(3) = 2 solutions are
  [(1,7,4),(3,9,6),(2,8,5)] and
  [(2,4,3),(6,8,7),(1,9,5)].

R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "I".
Peter Kagey, Haskell program for A279197.
Peter Kagey, Solutions for a(1)-a(10).
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.

A279199 Number of reducible ways to split 1, 2, 3, ..., 3n into n arithmetic progressions each with 3 terms: a(n) = A104429(n) - A202705(n).

Original entry on oeis.org

0, 0, 1, 3, 9, 30, 117, 512, 2597, 14892, 99034, 721350, 5909324, 52578654, 516148082, 5422071091, 61889692290, 749456672155, 9767058240577, 134007989313530, 1958535749524107

Offset: 0

N. J. A. Sloane, Dec 15 2016

R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "L".
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.

A002573 Restricted partitions.

Original entry on oeis.org

0, 1, 1, 2, 4, 7, 12, 22, 39, 70, 126, 225, 404, 725, 1299, 2331, 4182, 7501, 13458, 24145, 43316, 77715, 139430, 250152, 448808, 805222, 1444677, 2591958, 4650342, 8343380, 14969239, 26856992, 48185362, 86451602, 155106844, 278284440, 499283177, 895787396, 1607174300, 2883507098

Offset: 1

N. J. A. Sloane

Number of compositions n=p(1)+p(2)+...+p(m) with p(1)=2 and p(k) <= 2*p(k+1), see example. [Joerg Arndt, Dec 18 2012]

			From _Joerg Arndt_, Dec 18 2012: (Start)
There are a(8)=22 compositions 8=p(1)+p(2)+...+p(m) with p(1)=2 and p(k) <= 2*p(k+1):
[ 1]  [ 2 1 1 1 1 1 1 ]
[ 2]  [ 2 1 1 1 1 2 ]
[ 3]  [ 2 1 1 1 2 1 ]
[ 4]  [ 2 1 1 2 1 1 ]
[ 5]  [ 2 1 1 2 2 ]
[ 6]  [ 2 1 2 1 1 1 ]
[ 7]  [ 2 1 2 1 2 ]
[ 8]  [ 2 1 2 2 1 ]
[ 9]  [ 2 1 2 3 ]
[10]  [ 2 2 1 1 1 1 ]
[11]  [ 2 2 1 1 2 ]
[12]  [ 2 2 1 2 1 ]
[13]  [ 2 2 2 1 1 ]
[14]  [ 2 2 2 2 ]
[15]  [ 2 2 3 1 ]
[16]  [ 2 2 4 ]
[17]  [ 2 3 1 1 1 ]
[18]  [ 2 3 1 2 ]
[19]  [ 2 3 2 1 ]
[20]  [ 2 3 3 ]
[21]  [ 2 4 1 1 ]
[22]  [ 2 4 2 ]
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Shimon Even & Abraham Lempel, Generation and enumeration of all solutions of the characteristic sum condition, Information and Control 21 (1972), 476-482.
H. Minc, A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid, Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.

Cf. A002572, A047913, A002574, A049284, A049285.

A column of the triangle in A176431.

v := proc(c,d) option remember; local i; if d < 0 or c < 0 then 0 elif d = c then 1 else add(v(i,d-c),i=1..2*c); fi; end; [ seq(v(2,n), n=1..50) ];

v[c_, d_] :=  v[c, d] = If[d < 0 || c < 0, 0, If[d == c, 1, Sum[v[i, d - c], {i, 1, 2*c}]]]; a[n_] := v[2, n]; Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Jan 30 2012, after Maple *)

A194628 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 31, 61, 121, 240, 476, 944, 1872, 3712, 7362, 14601, 28958, 57432, 113904, 225904, 448034, 888583, 1762321, 3495200, 6932008, 13748208, 27266738, 54077957, 107252486, 212713209, 421872826, 836697836, 1659417786, 3291113315, 6527245245, 12945446241, 25674625681

Offset: 1

Jonathan Vos Post, Aug 30 2011

nonn

a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 5*p(k+1), see example. - Joerg Arndt, Dec 18 2012

Row 4 of Table 1 of Elsholtz, row 1 being A002572, row 2 being A176485, and row 3 being A176503.

			From _Joerg Arndt_, Dec 18 2012: (Start)
There are a(6+1)=16 compositions 6=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 5*p(k+1):
[ 1]  [ 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 2 ]
[ 3]  [ 1 1 1 2 1 ]
[ 4]  [ 1 1 1 3 ]
[ 5]  [ 1 1 2 1 1 ]
[ 6]  [ 1 1 2 2 ]
[ 7]  [ 1 1 3 1 ]
[ 8]  [ 1 1 4 ]
[ 9]  [ 1 2 1 1 1 ]
[10]  [ 1 2 1 2 ]
[11]  [ 1 2 2 1 ]
[12]  [ 1 2 3 ]
[13]  [ 1 3 1 1 ]
[14]  [ 1 3 2 ]
[15]  [ 1 4 1 ]
[16]  [ 1 5 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, The number of Huffman codes, compact trees, and sums of unit fractions, arXiv:1108.5964v1 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.

Cf. A002572, A176485, A176503, A294775.

b[n_, r_, k_] := b[n, r, k] = If[n < r, 0, If[r == 0, If[n == 0, 1, 0], Sum[b[n - j, k (r - j), k], {j, 0, Min[n, r]}]]];
a[n_] := b[4n - 3, 1, 5];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

PARI
```
/* see A002572, set t=5 */
```

a(n) = A294775(n-1,4). - Alois P. Heinz, Nov 08 2017

Terms beyond a(20)=113904 added by Joerg Arndt, Dec 18 2012

Invalid empirical g.f. removed by Alois P. Heinz, Nov 08 2017

A002574 Restricted partitions.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 7, 13, 24, 42, 76, 137, 245, 441, 792, 1420, 2550, 4576, 8209, 14732, 26433, 47424, 85092, 152670, 273914, 491453, 881744, 1581985, 2838333, 5092398, 9136528, 16392311, 29410243, 52766343, 94670652, 169853138, 304741614, 546751437, 980952673, 1759973660

Offset: 1

N. J. A. Sloane

Number of compositions n=p(1)+p(2)+...+p(m) with p(1)=3 and p(k) <= 2*p(k+1), see example. [Joerg Arndt, Dec 18 2012]

			From _Joerg Arndt_, Dec 18 2012: (Start)
There are a(8)=13 compositions 8=p(1)+p(2)+...+p(m) with p(1)=3 and p(k) <= 2*p(k+1):
[ 1]  [ 3 1 1 1 1 1 ]
[ 2]  [ 3 1 1 1 2 ]
[ 3]  [ 3 1 1 2 1 ]
[ 4]  [ 3 1 2 1 1 ]
[ 5]  [ 3 1 2 2 ]
[ 6]  [ 3 2 1 1 1 ]
[ 7]  [ 3 2 1 2 ]
[ 8]  [ 3 2 2 1 ]
[ 9]  [ 3 2 3 ]
[10]  [ 3 3 1 1 ]
[11]  [ 3 3 2 ]
[12]  [ 3 4 1 ]
[13]  [ 3 5 ]
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Shimon Even and Abraham Lempel, Generation and enumeration of all solutions of the characteristic sum condition, Information and Control 21 (1972), 476-482.
H. Minc, A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid, Proc. Edinburgh Math. Soc. (2) 11, 1958/1959, 223-224.

Cf. A002572, A002573, A049284, A049285, A047913.

v := proc(c,d) option remember; local i; if d < 0 or c < 0 then 0 elif d = c then 1 else add(v(i,d-c),i=1..2*c); fi; end; [ seq(v(3,n), n=1..50) ];

v[c_, d_] := v[c, d] = If[d < 0 || c < 0, 0, If[d == c, 1, Sum[v[i, d-c], {i, 1, 2*c}]]]; a[n_] := v[3, n]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Apr 05 2013, after Maple *)

More terms from Michael Somos

A047913 Triangle of numbers a(n,k) = number of partitions of k such that k = n + n_1 + n_2 + ... + n_t where n_1 <= 2n and n_{i+1} <= 2n_i for all i.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 4, 7, 9, 1, 1, 2, 4, 7, 12, 16, 1, 1, 2, 4, 7, 13, 22, 28, 1, 1, 2, 4, 7, 13, 24, 39, 50, 1, 1, 2, 4, 7, 13, 24, 42, 70, 89, 1, 1, 2, 4, 7, 13, 24, 43, 76, 126, 159, 1, 1, 2, 4, 7, 13, 24, 43, 78, 137, 225, 285

Offset: 1

N. J. A. Sloane

Triangle is read in this order: a(1,1), a(2,2), a(1,2), a(3,3), a(2,3), a(1,3), a(4,4), ...

Rows are the columns in the table at the end of the Minc reference, read bottom to top. - Joerg Arndt, Jan 15 2024

			Triangle begins:
1;
1, 1;
1, 1, 2;
1, 1, 2, 3;
1, 1, 2, 4, 5;
1, 1, 2, 4, 7,  9;
1, 1, 2, 4, 7, 12, 16;
1, 1, 2, 4, 7, 13, 22, 28;
1, 1, 2, 4, 7, 13, 24, 39, 50;
1, 1, 2, 4, 7, 13, 24, 42, 70,  89;
1, 1, 2, 4, 7, 13, 24, 43, 76, 126, 159;
1, 1, 2, 4, 7, 13, 24, 43, 78, 137, 225, 285;
...
Rows approach A002843. - _Joerg Arndt_, Jan 15 2024

H. Minc, A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid, Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.

Rows give A002572, A002573, A002574, ..., columns approach A002843.

Cf. A049286 (triangle with reversed rows).

a[n_, n_] = 1; a[n_?Positive, k_?Positive] := a[n, k] = Sum[a[i, k-n], {i, 1, 2*n}]; a[n_, k_] = 0; Table[a[n, k], {k, 1, 12}, {n, k, 1, -1}] // Flatten (* Jean-François Alcover, Oct 21 2013 *)

a(n, n)=1, a(n, k) = Sum_{i=1..2n} a(i, k-n).

cp's OEIS Frontend

A176485 First column of triangle in A176452.

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A176503 Leading column of triangle in A176463.

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A002848 Number of maximal collections of pairwise disjoint subsets {X,Y,Z} of {1, 2, ..., n} with X + Y = Z (as in A002849), with the property that n is in one of the subsets.

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A202705 Number of irreducible ways to split 1, 2, 3, ..., 3n into n arithmetic progressions each with 3 terms.

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A279197 Number of self-conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

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A279199 Number of reducible ways to split 1, 2, 3, ..., 3n into n arithmetic progressions each with 3 terms: a(n) = A104429(n) - A202705(n).

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A002573 Restricted partitions.

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A194628 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.